Generalisation of Levine's prediction for the distribution of freezing temperatures of droplets: a general singular model for ice nucleationR. P. SearDepartment of Physics, University of Surrey Guildford, Surrey GU2 7XH, UK

Abstract. Models without an explicit time dependence, called singular models,
are widely used for fitting the distribution of temperatures
at which water droplets freeze. In 1950 Levine
developed the original singular model. His key assumption
was that each droplet
contained many nucleation sites, and
that freezing occurred due to the nucleation site
with the highest freezing temperature.
The fact that freezing occurs due to the maximum value
out of a large number of nucleation temperatures, means
that we can apply the results of what is called extreme-value
statistics. This is the statistics of the extreme, i.e. maximum or
minimum, value of a large number of random variables.
Here we use the results of extreme-value statistics to show that
we can generalise Levine's model to produce
the most general singular model possible.
We show that when a singular model is a good approximation,
the distribution of freezing temperatures should always be given
by what is called the generalised extreme-value distribution.
In addition, we also show
that the distribution of freezing temperatures for droplets of one
size, can be used to make predictions for the scaling of
the median nucleation temperature with droplet size, and vice versa.